Geometry & Topology, Vol. 6 (2002)
Paper no. 18, pages 523--539.
Quantum $SU(2)$ faithfully detects mapping class groups modulo center
Michael H Freedman, Kevin Walker and Zhenghan Wang
Abstract.
The Jones-Witten theory gives rise to representations of the
(extended) mapping class group of any closed surface Y indexed by a
semi-simple Lie group G and a level k. In the case G=SU(2) these
representations (denoted V_A(Y)) have a particularly simple
description in terms of the Kauffman skein modules with parameter A a
primitive 4r-th root of unity (r=k+2). In each of these
representations (as well as the general G case), Dehn twists act as
transformations of finite order, so none represents the mapping class
group M(Y) faithfully. However, taken together, the quantum SU(2)
representations are faithful on non-central elements of M(Y). (Note
that M(Y) has non-trivial center only if Y is a sphere with 0, 1, or 2
punctures, a torus with 0, 1, or 2 punctures, or the closed surface of
genus = 2.) Specifically, for a non-central h in M(Y) there is an
r_0(h) such that if r>= r_0(h) and A is a primitive 4r-th root of
unity then h acts projectively nontrivially on V_A(Y). Jones' [J]
original representation rho_n of the braid groups B_n, sometimes
called the generic q-analog-SU(2)-representation, is not known to be
faithful. However, we show that any braid h not= id in B_n admits a
cabling c = c_1,...,c_n so that rho_N (c(h)) not= id, N=c_1 + ... +
c_n.
Keywords.
Quantum invariants, Jones-Witten theory, mapping class groups
AMS subject classification.
Primary: 57R56, 57M27.
Secondary: 14N35, 22E46, 53D45.
DOI: 10.2140/gt.2002.6.523
E-print: arXiv:math.GT/0209150
Submitted to GT on 14 September 2002.
Paper accepted 19 November 2002.
Paper published 23 November 2002.
Minor correction made 17 July 2003:
Sentence added to proof of lemma 4.1, page 536, lines 7-8.
Notes on file formats
Michael H Freedman, Kevin Walker and Zhenghan Wang
Microsoft Research, Redmond, WA 98052, USA (MHF and KW)
and
Indiana University, Department of Mathematics
Bloomington, IN 47405, USA
Email: michaelf@microsoft.com, kevin@messagetothefish.net, zhewang@indiana.edu
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